Matt, Scott:
There is a situation in one dimensional world, not so practically useful though, where the third cumulant is zero. It describes the 1st nearest neighbor interaction between a central atom (x) in the group of three atoms: A-----x-----A. Indeed, here the third cumulant is zero, the higher order - may be not, depending on the shape of the pair potential. Assume the pair potential as:
V(x) = 1/2*kx^2 + k3*x^3, (1)
Then, the asymmetric term vanishes since:
V_eff(x) = V(x) + V(-x) = kx^2. (2)
It is not obvious that there exist other examples. As I am sure you know, Hung, Hung/Rehr, Yokoyama, Pirog and others have studied behavior of cumulants in the most common structures, including fcc, hcp and bcc, using Morse potential. In all of them, the third cumulants were nonzero (albeit Hung/Rehr article contained an error in the third cumulant calculation, and these results were further generalized by Vila/Rehr in their recent paper.
Indeed, the potential in these cases is always a linear combination of crystallography-defined terms, e.g., for the fcc structure, assuming a displacement x along the nearest neighbor distance:
V_eff(x) = V(x)+2V(-x/2)+8V(x/4)+8V(-x/4)+4V(0) = 1/2*(5/2)*kx^2 + (3/4)*k3*x^3. (3)
If you take above equation for V(x) and plug it in, you will obtain that V_eff(x) is still anharmonic since the x^3 term is still present in the effective contribution. It does not matter in this case, whether the atoms surrounding the dopant are "fixed" or "not fixed", as long as their is an effective pair potential between the central atom and its neighbors described by equation (1).
Same results can be obtained for most other lattices. Note that even if for some displacement direction x the sum of the terms in Eq. (3) will have no x^3, there is no chance that it will be true for ALL directions, since there is nothing special in x pointing along the 1NN distance.
Thus, I am pretty much convinced, unless there is some mistake in my reasoning, that no case exists in 3D with a zero third cumulant.
Anatoly
________________________________
From: ifeffit-bounces@millenia.cars.aps.anl.gov on behalf of Matt Newville
Sent: Wed 1/21/2009 1:27 PM
To: XAFS Analysis using Ifeffit
Subject: Re: [Ifeffit] Cumulant expansion fittings
Umesh, Scott, Anatoly,
I'm not sure what Scott means by a "lattice atom" in "the limit in which
the lattice atoms are fixed in place". Materials have atoms. Crystals and
lattice points. But I think I do agree with his approach to constructing a
non-harmonic distribution for which the third cumulant is zero.
Cumulants are simple combinations of the moments of a distribution that are
particularly useful when one has an exponential function of a variable and
wants to model a distribution function of that variable. For XAFS, the
distribution we care about is r (interatomic distance), which has moments:
/
On Jan 21, 2009, at 10:12 AM, Frenkel, Anatoly wrote:
Hi Scott,
Third cumulant in your example will not be zero because this arrangement is symmetric only on the average. Locally, the interatomic pair potential (and the cumulants are the measures of the effective pair potential) which is the sum of the two potentials - between the interestitial and its neighbors on the opposite sides)- is still asymmetric, since the repulsive bruch of the potential is steeper than the attractive brunch. You can model your situation using two anharmonic pair potentials, e.g., Morse potential (see, for example, Rehr-Hung's paper in Phys Rev B in the 1990's, and I've done such calculations too, just in the case you described) and you will obtain that the effective pair potential is still analytically anharmonic and it has a non-zero third cumulant.
Anatoly
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